It is the conformal automorphism group of the Macbeath surface, which has genus 7, and the order of the group is , so it is a Hurwitz group.

GAP implementation

Group ID

This finite group has order 504 and has ID 156 among the groups of order 504 in GAP's SmallGroup library. For context, there are 202 groups of order 504. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(504,156)

For instance, we can use the following assignment in GAP to create the group and name it :

gap> G := SmallGroup(504,156);

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [504,156]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.